Mod-two handle duality ($H_k(W;{\mathbb{F}}_2)\cong H^{n-k}(W,\partial W;{\mathbb{F}}_2)$)

Reverse a handle decomposition of a compact $n$-dimensional manifold. Absolute $k$-handles become relative $(n-k)$-handles based on the boundary. The absolute boundary matrices count attaching sphere-belt sphere intersections; the corresponding relative matrices are their transposes over ${\mathbb{F}}_2$. The absolute chain complex is thus the complementary-degree dual of the relative chain complex. The universal coefficient theorem for cohomology proves Poincare-Lefschetz duality without an orientation hypothesis.